There are \(5\) directors of \(5\) banks sitting at the round table. Some of these banks have a negative balance (they owe more money that they have) and some have a positive balance (they have more money that they owe). It is known that for any 3 directors sitting next to each otehr, their 3 banks together have a positive balance. Does it mean that the \(5\) banks together have a positive balance?
A group of Martians and a group of Venusians got together for an important talk. At the start of the meeting, each Martian shook hands with 6 different Venusians, and each Venusian shook hands with 8 different Martians. It is known that 24 Martians took part in the meeting. How large was the delegation for Venus?
There are \(36\) warrior tomcats standing in a \(6 \times 6\) square formation. Each cat has several daggers strapped to his belt. Is it possible that the total number of daggers in each row is more than \(50\) and the total number of daggers in each column is less than \(50\)?
Francesca, Isabella and Lorenzo played chess together. Each child played \(10\) rounds.
a) What was the total number of rounds?
b) Is it possible that Lorenzo played more rounds with Isabella than with Francesca?
A \(3 \times 3\) magic square is a square with different number from \(1\) to \(9\) in each of its \(9\) cells. The numbers in each row, column and diagonal sum up to \(15\). Show that there is a number \(5\) in the centre of the square.
Can you decorate an \(8 \times 8\) cake with chocolate roses in such a way that any \(2 \times 2\) piece would have exactly 2 roses on it, and any \(3 \times 1\) piece would have exactly one rose? Either draw such a cake or explain why this is not possible.
Several films were nominated for the “Best Math Movie“ award. Each of the 10 judges secretly picked the top movie of their choice. It is known that out of any 4 judges, at least 2 voted for the same film. Prove that there exists a film that was picked by at least 4 judges.
Out of \(7\) integer numbers, the sum of any \(6\) is a multiple of \(5\). Show that every one of these numbers is a multiple of \(5\).
An \(8 \times 8\) chessboard has 30 diagonals total (15 in each direction). Is it possible to place several chess pieces on this chessboard in such a way that the total number of pieces on each diagonal would be odd?
Anna’s garden is a grid of \(n \times m\) squares. She wants to have trees in some of these squares, but she wants the total number of trees in each column and in each row to be an odd number (not necessarily the same, they just all need to be odd). Show that it is possible only if \(m\) and \(n\) are both even or both odd and calculate in how many different ways she can place the trees in the grid.